Cycle graph:C5

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
View a complete list of particular undirected graphs

Definition

This undirected graph is defined in the following equivalent ways:

1. It is the cycle graph on 5 vertices, i.e., the graph $C_{5}$
2. It is the Paley graph $P_{5}$ corresponding to the field of 5 elements
3. It is the unique (up to graph isomorphism) self-complementary graph on a set of 5 vertices

Note that 5 is the only size for which the Paley graph coincides with the cycle graph. In general, the Paley graph $P_{q}$ can be expressed as an edge-disjoint union of $(q-1)/4$ cycle graphs. In our case, $(5-1)/4=1$, so the graphs coincide.

Explicit descriptions

Descriptions of vertex set and edge set

Vertex set: $\{1,2,3,4,5\}$

Edge set: $\{\{1,2\},\{2,3\},\{3,4\},\{4,5\},\{1,5\}\}$

Adjacency matrix

The adjacency matrix with the above choice of vertex set is:

${\begin{pmatrix}0&1&0&0&1\\1&0&1&0&0\\0&1&0&1&0\\0&0&1&0&1\\1&0&0&1&0\\\end{pmatrix}}$

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	5	By either definition
size of edge set	5	As $C_{n},n=5$: $n=5$ As $P_{q},q=5$: $q(q-1)/4=5(4)/4=5$

Numerical invariants associated with vertices

Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:

Function	Value	Explanation
degree of a vertex	2	As $C_{n},n=5(n\geq 3)$: 2 As $P_{q},q=5$: $(q-1)/2=(5-1)/2=2$
eccentricity of a vertex	2	As $C_{n},n=5(n\geq 3)$: greatest integer of $n/2$ equals 2 As $P_{q},q=5$: 2 (follows from every element of a finite field is expressible as a sum of two squares)

Other numerical invariants

Function	Value	Explanation
clique number	2	As cycle graph $C_{n},n=5$: 2 (since $n\geq 4$)
independence number	2	As cycle graph $C_{n},n=5$: Greatest integer of $n/2$, which is greatest integer of $5/2$, which is 2
chromatic number	3	As cycle graph $C_{n},n=5$: 3 since $n$ is odd
radius of a graph	2	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	2	Due to vertex-transitivity, the diameter equals the eccentricity of any vertex, which has been computed above.
girth of a graph	5	As $C_{n},n=5$: $n=5$ As $P_{q},q=5$: 5. Note that for all higher $q$, the girth of the graph is 3, due to the fact that there are $(q-5)/4$ pairs of nonzero quadratic residues that differ by 1, and this number is positive whenever $q>5$.
circuit rank	1	As $C_{n},n=5$: 1 (doesn't depend on $n$) As $P_{q},q=5$: $q(q-5)/4+1=5(5-5)/4+1=1$

Graph properties

Property	Satisfied?	Explanation
self-complementary graph	Yes	Paley graphs are self-complementary
strongly regular graph	Yes	Paley graphs are strongly regular
regular graph	Yes	Follows from being strongly regular. Also follows from the fact that cycle graphs are 2-regular.
conference graph	Yes	Paley graphs are conference graphs
symmetric graph	Yes
edge-transitive graph	Yes	Follows on account of being Paley and also on account of being cyclic
vertex-transitive graph	Yes

Graph operations

Operation	Graph obtained as a result of the operation
complement of a graph	isomorphic to the cycle graph, because the graph is self-complementary
line graph	complement of Petersen graph
prism of a graph	dihedral graph on 10 vertices

Algebraic theory

The adjacency matrix, well defined up to conjugation by permutations, is:

${\begin{pmatrix}0&1&0&0&1\\1&0&1&0&0\\0&1&0&1&0\\0&0&1&0&1\\1&0&0&1&0\\\end{pmatrix}}$

Algebraic invariant	Value	Explanation
characteristic polynomial	$t^{5}-5t^{3}+5t-2=(t^{2}+t-1)^{2}(t-2)$
minimal polynomial	$t^{3}-t^{2}-3t+2=(t^{2}+t-1)(t-2)$
rank of adjacency matrix	3
eigenvalues (roots of characteristic polynomial)	2 (1 time), ${\frac {-1+{\sqrt {5}}}{2}}$ (2 times), ${\frac {-1-{\sqrt {5}}}{2}}$ (2 times)

Realization

As Cayley graph

Note that for this to be the Cayley graph of a group, the group must have order 5, and the generating set with respect to which we construct the Cayley graph must be a symmetric subset of the group of size equal to the degrees of vertices in the graph, which is 2.

Group	Choice of symmetric set that is a generating set for which the Cayley graph is this
cyclic group:Z5	cyclic generator and its inverse